The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.

The figure is divided into 6 sectors by the blue trunks. In the interior of each sector, working outwards from the base point P at the origin, there are successively 1,2,3,4,... (red) 12-gons. All the 12-gons (both red and blue) have a unique closest point to P.

If the closest point in a 12-gon is at distance d from P, then the contributions of the 12 points of the 12-gon to a(d), a(d+1), ..., a(d+6) are 1,2,2,2,2,2,1, respectively.

The rest of the proof is now a matter of simple counting.

The blue 12-gons (along the trunks) are especially easy to count, because there is a unique blue 12-gon at shortest distance d from P for d = 1,2,3,4,...